Possibly reflecting British slang for "immature weakling" implying that intellectual "weenies" are piling onto Cheney's decline in power with pondering analysis.

On "Meet the Press", 9/10/06, host Tim Russert questioned the Vice President about an article in the New York Times published on the same day entitled "Cheney’s Power No Longer Goes Unquestioned" by David E. Sanger and Eric Schmitt.

(Admittedly the specific meaning needs pegging better, but hey this is a start.)

"...that (New York Times article) is just another one of those thumbsuckers..." -- Richard "Deadeye Dick" Cheney (Sept. 10, 2006)

An adult who whines about very trivial things. Making them appear childish.

A woman sitting at a traffic light, behind a car displaying a non-Christian Evolve Fish emblem, yelled unchristian obscenities to the driver.
The passenger in the car next to her rolled down his window, and calmly asked? "Do you realize how childish your behaving?"
At this she screamed more angry obscenities.
The passenger calmly waited for her to calm down, and as if speaking to a child, said. "Where's your mommy? Thumbsucker."